The quantum numbers n and l for four electrons are given below. (i) n = 4, I = 1 (ii) n = 4, l = 0 (iii) n = 3, l = 2 (iv) n = 3, l = 1 The order of their energy from lowest to highest is:
Determining Electron Energy Order using Quantum Numbers (n and l)
The energy of an electron in a multi-electron atom is primarily determined by the principal quantum number (n) and the azimuthal or angular momentum quantum number (l). According to the (n+l) rule, also known as the Bohr-Bury rule, the orbital with the lower value of (n+l) has lower energy.
If two different orbitals have the same (n+l) value, the orbital with the lower value of 'n' has lower energy. This rule helps us determine the filling order of orbitals and compare the energy levels of electrons within different orbitals.
Let's examine the given quantum numbers for the four electrons:
- (i) n = 4, l = 1
- (ii) n = 4, l = 0
- (iii) n = 3, l = 2
- (iv) n = 3, l = 1
Now, we calculate the value of (n+l) for each electron:
| Electron | n | l | n+l |
|---|---|---|---|
| (i) | 4 | 1 | \(4+1 = 5\) |
| (ii) | 4 | 0 | \(4+0 = 4\) |
| (iii) | 3 | 2 | \(3+2 = 5\) |
| (iv) | 3 | 1 | \(3+1 = 4\) |
Based on the (n+l) values, we can see two groups:
- (ii) and (iv) have \(n+l = 4\).
- (i) and (iii) have \(n+l = 5\).
Orbitals with \(n+l=4\) are lower in energy than those with \(n+l=5\). So, the order will be something like \((iv), (ii)\) < \((i), (iii)\).
Now, we apply the tie-breaking rule for the orbitals with the same (n+l) value:
- For (ii) and (iv), both have \(n+l=4\). We compare their 'n' values:
- (ii) has n = 4
- (iv) has n = 3
- For (i) and (iii), both have \(n+l=5\). We compare their 'n' values:
- (i) has n = 4
- (iii) has n = 3
Combining these comparisons, the overall order of energy from lowest to highest is:
Electron (iv) (n=3, l=1, n+l=4) < Electron (ii) (n=4, l=0, n+l=4) < Electron (iii) (n=3, l=2, n+l=5) < Electron (i) (n=4, l=1, n+l=5).
Thus, the order is (iv) < (ii) < (iii) < (i).
Revision Table: Quantum Numbers and Orbital Energy
| Quantum Number | Symbol | Describes | Allowed Values |
|---|---|---|---|
| Principal Quantum Number | n | Energy level (shell), size of orbital | 1, 2, 3, ... (positive integers) |
| Azimuthal (Angular Momentum) Quantum Number | l | Shape of orbital (subshell) | 0, 1, 2, ..., n-1 |
| Magnetic Quantum Number | m\(_l\) | Orientation of orbital in space | -l, -l+1, ..., 0, ..., l-1, l |
| Spin Quantum Number | m\(_s\) | Spin of the electron | +1/2 or -1/2 |
Additional Information on Electron Configuration and Energy Levels
The relative energies of orbitals in a multi-electron atom determine how electrons fill these orbitals according to the Aufbau principle, Hund's rule, and the Pauli exclusion principle.
- Aufbau Principle: Electrons fill orbitals in order of increasing energy. The (n+l) rule helps predict this order.
- Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers. This means an atomic orbital can hold a maximum of two electrons, and these two electrons must have opposite spins.
- Hund's Rule: For orbitals within the same subshell (same n and l, but different m\(_l\)), electrons fill each orbital singly with parallel spins before pairing up in any one orbital.
The energy levels of orbitals are not fixed but can be affected by electron-electron repulsions and shielding effects, especially in larger atoms. However, the (n+l) rule provides a very useful and generally accurate guideline for predicting the relative energies and filling order of orbitals.
Understanding quantum numbers and energy levels is fundamental to understanding atomic structure, chemical bonding, and the properties of elements.
Important Questions from Atomic Structure
The four quantum numbers of the valence electron of potassium atom are:
An element with atomic number 103 has a nuclear charge of +103. So its charge would be balanced by ________.
An electron near the nucleus is strongly attracted by the nucleus and has _________.
How many emission spectrum for sodiumlie in the range 588 - 590 nm?
What is the atomic number of Bohrium which is named after physicist Niels Bohr, one of the founders of quantum theory?
